On families of quadratic surfaces having fixed intersections with two hyperplanes

In this paper we investigate families of quadrics that have fixed intersections with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter. In particular we show how the quadrics are transformed as the parameter changes. This research was motivated by an application in mixed-integer conic optimization. In that application we aimed to characterize the convex hull of the union of the intersections of an ellipsoid with two half-spaces when these intersections are disjunctive sets.